How to join, and intersect objects?
Given that the two edges intersect:




Select edge1 and create a new Transform Orientation from the edge's normal orientation.
Select edge2 and snap the cursor to it. &#8679; ShiftS > Cursor to Selected
Subdivide edge1 and select the created vertice. W > S
Set the Pivot center for rotation/scaling to 3D Cursor with ..
Scale the vertice to 0 along its local Y axis. SYY0





Delete edge2 (X) and connect the vertices (F).